Properties

Label 4-20e4-1.1-c2e2-0-10
Degree $4$
Conductor $160000$
Sign $1$
Analytic cond. $118.792$
Root an. cond. $3.30139$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·9-s + 40·13-s + 30·17-s − 96·29-s + 20·37-s + 66·41-s + 38·49-s + 60·53-s + 76·61-s + 10·73-s − 72·81-s + 174·89-s + 220·97-s + 84·101-s + 116·109-s − 270·113-s + 120·117-s − 133·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 90·153-s + 157-s + 163-s + ⋯
L(s)  = 1  + 1/3·9-s + 3.07·13-s + 1.76·17-s − 3.31·29-s + 0.540·37-s + 1.60·41-s + 0.775·49-s + 1.13·53-s + 1.24·61-s + 0.136·73-s − 8/9·81-s + 1.95·89-s + 2.26·97-s + 0.831·101-s + 1.06·109-s − 2.38·113-s + 1.02·117-s − 1.09·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.588·153-s + 0.00636·157-s + 0.00613·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(160000\)    =    \(2^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(118.792\)
Root analytic conductor: \(3.30139\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 160000,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.122031988\)
\(L(\frac12)\) \(\approx\) \(3.122031988\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 - p T^{2} + p^{4} T^{4} \)
7$C_2^2$ \( 1 - 38 T^{2} + p^{4} T^{4} \)
11$C_2^2$ \( 1 + 133 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 - 20 T + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 - 15 T + p^{2} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 347 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - 998 T^{2} + p^{4} T^{4} \)
29$C_2$ \( ( 1 + 48 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 422 T^{2} + p^{4} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
41$C_2$ \( ( 1 - 33 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 142 T^{2} + p^{4} T^{4} \)
47$C_2^2$ \( 1 - 4178 T^{2} + p^{4} T^{4} \)
53$C_2$ \( ( 1 - 30 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 962 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 38 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 5603 T^{2} + p^{4} T^{4} \)
71$C_2^2$ \( 1 - 4082 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 5 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 10982 T^{2} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 13643 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 87 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 110 T + p^{2} T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.38598293421377270884312308899, −10.90375144662421179950338224654, −10.38184202629161785696846275292, −10.05505072262802189813918027094, −9.226871944569710794836071996331, −9.152626124365982735999993786884, −8.576920510876075282742797181042, −8.041534121455637884932573117941, −7.43784801091939358315347428271, −7.38668775027577060799978827986, −6.28011311401307742568984895775, −6.08430346717982830063534151215, −5.61908998079389097291841634937, −5.15430097735605721649093919441, −4.01205280384165766938305139596, −3.75832674843902557614065210586, −3.46988790842823355144371868733, −2.34714287017996258057256593177, −1.42596040120943291196166209147, −0.893718891552934935462154322844, 0.893718891552934935462154322844, 1.42596040120943291196166209147, 2.34714287017996258057256593177, 3.46988790842823355144371868733, 3.75832674843902557614065210586, 4.01205280384165766938305139596, 5.15430097735605721649093919441, 5.61908998079389097291841634937, 6.08430346717982830063534151215, 6.28011311401307742568984895775, 7.38668775027577060799978827986, 7.43784801091939358315347428271, 8.041534121455637884932573117941, 8.576920510876075282742797181042, 9.152626124365982735999993786884, 9.226871944569710794836071996331, 10.05505072262802189813918027094, 10.38184202629161785696846275292, 10.90375144662421179950338224654, 11.38598293421377270884312308899

Graph of the $Z$-function along the critical line